Geodesic manifold

Examples

All compact Riemannian manifolds and all homogeneous manifolds are geodesically complete.

Euclidean space R n , the spheres S n and the tori T n (with their natural Riemannian metrics) are all complete manifolds.

A simple example of a non-complete manifold is given by the punctured plane M := R 2 ∖ { 0 } (with its induced metric). Geodesics going to the origin cannot be defined on the entire real line.

There exists non geodesically complete compact pseudo-Riemannian (but not Riemannian) manifolds. It is the case for example of the Clifton–Pohl torus.

Path-connectedness, completeness and geodesic completeness

It can be shown that a finite-dimensional path-connected Riemannian manifold is a complete metric space (with respect to the Riemannian distance) if and only if it is geodesically complete. This is the Hopf–Rinow theorem. This theorem does not hold for infinite-dimensional manifolds. The example of a non-complete manifold (the punctured plane) given above fails to be geodesically complete because, although it is path-connected, it is not a complete metric space: any sequence in the plane converging to the origin is a non-converging Cauchy sequence in the punctured plane.